Let S_k, k=1,2, , 100 , denote the sum of the infinite geometric series whose first term is k-1 k ! and the common ratio is 1 k . Then the value of 100^2 100 ! + _ k=1 ¹⁰⁰ | (k^2-3 k+1 ) S_k | is

Step-by-step solution

We have, S_k= k-1 k ! 1- 1 k = 1 (k-1) ! Now, aligned & (k^2-3 k+1 ) S_k= (k-2)(k-1)-1 & = 1 (k-3) ! - 1 (k-1) ! & _ k=1 ¹⁰⁰ | (k^2-3 k+1 ) S_k | & =1+1+2- ( 1 99 ! + 1 98 ! )=4- 100^2 100 ! aligned 100^2 100 ! + _ k=1 ¹⁰⁰ | (k^2-3 k+1 ) S_k |=4